Abstract
Variational method is a useful mathematical tool in various areas of research, espe-
cially in image processing. Recently, solving image processing problems on general manifolds using
variational techniques has become an important research topic. In this paper, we solve several image
processing problems on general Riemann surfaces using variational models defined on surfaces. We
propose an explicit method to solve variational problems on general Riemann surfaces, using the
conformal parameterization and covariant derivatives defined on the surface. To simplify the com-
putation, the surface is firstly mapped conformally to the two dimensional rectangular domains, by
computing the holomorphic 1-form on the surface. It is well known that the Jacobian of a conformal
map is simply the scalar multiplication of the conformal factor. Therefore with the conformal para-
meterization, the covariant derivatives on the parameter domain are similar to the usual Euclidean
differential operators, except for the scalar multiplication. As a result, any variational problem on
the surface can be formulated to a 2D problem with a simple formula and efficiently solved by well
developed numerical scheme on the 2D domain. With the proposed method, we solve various image
processing problems on surfaces using different variational models, which include image segmen-
tation, surface denoising, surface inpainting, texture extraction and automatic landmark tracking.
Experimental results show that our method can effectively solve variational problems and tackle
image processing tasks on general Riemann surfaces.